It is the interaction between the frequencies of different notes played together that gives music its richness and color. These notes played together are called Chords.

As there are rules in their construction, let's see how chords are constructed from the harmonies made from playing several notes together.
 

1. Harmony :

Playing more than one note at the same time is called harmony.  The difference in the pitch between two notes is called the interval.

There are many different interval sizes, from the smallest interval of one semitone up to an octave and beyond.  Each interval size has a different sound, and is given its own name based on the notes in the Major scale.

The richness and variety of harmony multiplies with each note you add. Harmonies with three or more notes are called chords and they provide the harmonic structure or background mood of a piece of music.  Intervals are the building blocks of chords.
     

2. Interval Sizes :

The sound and feel of different interval sizes varies greatly.  With some intervals, notes blend naturally together to create a pleasing (consonant) sound.  Other intervals create a more complex and colored (dissonant) sound.

Both consonant and dissonant intervals are used in music, in combination.  Too much consonance in music makes it easy to listen to.(Commonly used in Pop Music, Ballads, Rock, Heavy Metal...etc)

Dissonance adds a tension to music which, if used well, has a real "powerful" effect.  Too much dissonance, makes music less accessible, except to a narrower audience. (Dissonance chords are widely used in Jazz, Bossa Nova, Jazz-Rock...)

3. Shown below are all the intervals and their sizes (measured in semitones) of up to an octave, with a brief description of their sound.

1. Unison intervals are two identical notes played together.  They are always strongly consonant, and difficult to tell apart.
 

2. Minor second intervals are strongly dissonant, with a warbling sound in the background, as if the two notes are fighting with each other. Major Second intervals are less dissonant, but the notes still do not sit completely at ease with each other. 
 
3. Minor third intervals are strongly consonant, with a melancholy flavour to the sound.  They form the basis of minor chords and scales. Major third intervals are strongly consonant, making a stable and pleasing sound.  They forms the basis of major chords / scales. 
 
4. Perfect fourth intervals are mildly dissonant, with a stretched feeling as if it would rather return to a major third. 
 Tritone intervals are dissonant, and are often found in chords of four notes or more, where it adds a particular harmonic spice. 
 

5. Perfect fifth intervals are strongly consonant, and are found in both minor and major chords.  They add solidness, but not much character to the harmony.
 
6. Minor sixth intervals are mildly dissonant.  Major sixth intervals are consonant (used widely in Jazz, Bossa Nova & Fusion).

7. Minor seventh intervals are mildly dissonant. Major seventh intervals are dissonant (used widely in Blues)
 
8. Octave intervals are strongly consonant, like unison, because notes an octave apart sound similar to each other, just higher or lower. 
 

Intervals of more than twelve semitones are repeats of these intervals, but with a slightly different effect because of the extra octave of separation.  Intervals of up to 21 semitones are regularly used in the construction of chords. These intervalls are used in Jazz and Latin music mostly but also in Arabic and Indian musics.
 

 
4. Basic chords, practically :

Beyond theory about intervalls and chords - on which it might possible to write a book -, let's focus on the commonly used Chords : They are usually composed of 3 basic notes (also called Triad):

• a tonic / root (=the note who give its name to the Chord)

• a "third" (note that minor third will sound a bit melancholic while a major third will sound more positively entertaining)

• a fifth.

Example : the Chord "C Major" will be made of C (tonic), E (major third), G (fifth)

5. About Major Chord Types & harmonic extensions :

Chords being built from the combination of intervals formed by playing three (Triad) or more different notes together, there are many different chord types, and each one is defined by its formula, a unique combination of intervals.  Each of these intervals is measured from the starting tonic note.

In the above example above, the major chord has a formula made from the triad Tonic + third + fifth (also called degrees 1, 3 and 5)

 
Extended chords are the chord types of four or more notes which are formed by adding extra notes to a triad.  There are many ways to extend each of the triads, depending on the combination of notes added.

Note that this series contains the flat seventh degree (also called dominant seventh) rather than the perfect seventh.  With chords, the flat seventh is so common that it is simply called the seventh, and the perfect seventh is referred to as the major seventh to distinguish it.

Here are the chord types formed by the harmonic extension of the major triad.
 
 

Note : b7 means the seventh is a minor one (10 semitones)

Other Major Chords : Many other chord types can be formed from the major triad (1, 3, 5), using other degrees, or a combination of harmonic and other degrees.  These are listed below.
 
 

 
6. About Minor Chord Types & harmonic extensions :

As major chords are made of a major third, there is a serie of chord types formed by the harmonic extension of the minor triad.  The only difference between Major and Minor Chord is their third (which gives the melancholic or happy "color")

Previous example was : Chord "C Major" will be made of C (tonic), E (major third), G (fifth)

Current example was : Chord "C Minor" will be made of C (tonic), D#(minor third), G (fifth)

Therefore, all chords structure listed in the previous tables above can be build as well , just playing a Minor third (3 semitones) instead of a Major third (4 semitones). This also applied for :

7. Major Flat Fifth Chords :

Several of these are similar to major chords, with the only difference of a one semitone diminished fifth degree.
   

8. Augmented Chords :

These often correspond to the Major Flat Fifth chord types listed above.  The symbols "#5" and "+" in chord names both indicate the augmenting of the fifth degree.
 

9. Naming some more Chords :

You can make new chord names from existing ones by using the original name and manipulating the degrees of it.  For example, you may like the sound of the chord which is a m11 (minor eleventh) but with the fifth degree raised by a semitone (augmented).

This particular chord has not been listed above, but when you call it "m11#5" (or minor 11 and augmented 5th), it is readily understood by other musicians that might play with you or it is just a convenient way to remember them during your compositions for example.